Linking structure and activity in nonlinear spiking networks

Recent experimental advances are producing an avalanche of data on both neural connectivity and neural activity. To take full advantage of these two emerging datasets we need a framework that links them, revealing how collective neural activity arises from the structure of neural connectivity and intrinsic neural dynamics. This problem of structure-driven activity has drawn major interest in computational neuroscience. Existing methods for relating activity and architecture in spiking networks rely on linearizing activity around a central operating point and thus fail to capture the nonlinear responses of individual neurons that are the hallmark of neural information processing. Here, we overcome this limitation and present a new relationship between connectivity and activity in networks of nonlinear spiking neurons by developing a diagrammatic fluctuation expansion based on statistical field theory. We explicitly show how recurrent network structure produces pairwise and higher-order correlated activity, and how nonlinearities impact the networks’ spiking activity. Our findings open new avenues to investigating how single-neuron nonlinearities—including those of different cell types—combine with connectivity to shape population activity and function.Author summary: Neuronal networks, like many biological systems, exhibit variable activity. This activity is shaped by both the underlying biology of the component neurons and the structure of their interactions. How can we combine knowledge of these two things—that is, models of individual neurons and of their interactions—to predict the statistics of single- and multi-neuron activity? Current approaches rely on linearizing neural activity around a stationary state. In the face of neural nonlinearities, however, these linear methods can fail to predict spiking statistics and even fail to correctly predict whether activity is stable or pathological. Here, we show how to calculate any spike train cumulant in a broad class of models, while systematically accounting for nonlinear effects. We then study a fundamental effect of nonlinear input-rate transfer–coupling between different orders of spiking statistic–and how this depends on single-neuron and network properties.